1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. THE LIMIT MANIFOLD (M;;,,,g 00 ) 173

Similarly, we have

again since Cr 2: 1. Hence, given E > 0 and p > 0, we can take j 0 ....:...

max (1 - log 2 E,p).

For j = 0, make ko large enough so that Fko£;l is a ( C 0 1, 0)-approximate

isometry for any .e 2: ko (we can do this by Proposition 4.33). By induction,

suppose we can do this up to kr. We then make kr+l large enough so that

Fk,+i£;2r+i is a ( c;i 1 2-(r+l), r + 1 )-approximate isometry for any .e 2: kr+l ·

Now choose Wr ~ Fkrkr+i;2r. D

Let us re-index the subsequence taken in the previous proposition so

that it is once again indexed by k and the index of w coincides with the

index of M. We may now take the final subsequence to get metrics on

B (Ok, 2k) c Mk which will become the limit metric. Since Wj,£ are ap-

proximate isometries, we may consider the ~equence {wj e9j+e}= of Rie-

, £=0
mannian metrics on B ( Oj, 2j). Since Wj,£ are (E,p)-approximate isometries
independent of .e, the.metrics wj,e9j+e are uniformly bounded together with
its derivatives and so there is a subsequence in .e so that they converge to a
limit metric 9j,oo on B ( Oj, 2j). We can use this argument diagonally to get
the following pro:position. ·

PROPOSITION 4.40 (Existence of almost-isometric limiting metrics on
large balls). There exist a subsequence { kj} ~ 1 and Riemannian metrics

9kj,oo on B ( Okj, 2kj) such that for every E > 0 and p 2: 0 there exists

jo = jo (E,p) such that · ·

for all r :=::; p and .e 2: 0 if j 2: jo.

PROOF. This essentially follows from Lemma 4.37 again. D

We again re-index, replacing wkj with Wj, which equals wkHi o · · · o
wk·+2 J 0 wk·+l J 0 wk· J in the old notation, so that we have a sequence of maps
Wj: B (Oj,2j) _, B (Oj+l,2j+l) (note that if j corresponds to kj, then we
have shrunk the ball of radius 2kj to the ball of radius 2j).

We note that Wj: (B (Oj,2j) ,gj,oo) -t (B (Oj+l,2j+l) ,gj+1, 00 ) is an

isometry since

lw.fgj+l,oo - 9j,ool ::::; lw; (9j+l,oo - w;+l · · · w.f+e-19j+e) I

+ lwjwj+l · · · w.f+.e-19H.e - 9j,ool

and both terms on the RHS go to zero as .e -t oo.